2015 Students

Name (Last, First)	Home Institution	Country
Bohdan, Kivva	Taras Shevchenko National University of Kyiv	Ukraine
Celebi, Recep	Lafayette College	U.S.A.
Chaudkhari, Maksym	Taras Shevchenko National University of Kyiv	Ukraine
Chen, Yichao	University of Toronto	Canada
Gelwan, Noah	University of Rochester	U.S.A.
Guo, Zhixing	University of California, Santa Barbara	U.S.A.
Harris, Connor	Harvard University	U.S.A.
He, Changhan	University of Science and Technology of China	China
Hendricks, Kirk	University of Arizona	U.S.A.
Ibarra Rodriguez, José	University of Guanajuato	Mexico
Jordan, Matthew	McMaster University	Canada
Khater, Nada	McMaster University	Canada
Kim, Hwanwoo	University of Michigan	U.S.A.
Ladtchenko, Vsevolod	University of Toronto	Canada
Li, Qian	University of Toronto	Canada
Morgenstern, Ilan	Instituto Tecnológico Autónomo de México	Mexico
Reynolds, Michael	Trinity College Dublin, The University of Dublin	Ireland
Rudenko, Oleksandr	Taras Shevchenko National University of Kyiv	Ukraine
Shan, Bingchen	University of Toronto	Canada
Tong, Ke	University of Toronto	Canada
Viger, Torin	University of Toronto	Canada
Zhang, Hongjing	University of Toronto	Canada

 

Application Process

Please apply for the 2015 Program here.

During the application process, you will be prompted to upload your Curriculum Vitae (CV) together with a letter outlining your relevant background and experience. Your letter must not exceed two letter-sized pages, and must be in 12-point Times New Roman font, with single line spacing. Top, bottom, and side margins each must be no less than one inch. Your letter and CV must be submitted in a single, combined PDF document.

For your application to be considered, you must also arrange to have the following documents provided directly to the Fields Institute, on or before the February 28, 2015 application deadline noted below:

1. Official transcripts* from your home university addressed to:

Fields Undergraduate Summer Research Program
Fields Institute for Research in Mathematical Sciences
222 College Street
Toronto, ON
M5T 3J1 Canada

*An official transcript is prepared and sent by the issuing school usually by the Student Registrar with an original signature of a school official.

2. Two confidential letters of reference from someone who can provide a candid evaluation of your qualifications or skills. These should be sent by the referee, in confidence, directly to the Fields Institute, either by email from the referee's email account to programs@fields.utoronto.ca, or by mail or courier to the address noted above.

To be considered for the Program, all of your materials must be received by the Fields Institute before 11:59 pm Eastern Standard Time on February 28, 2015.

Accepted students requiring visas for travel to Canada will need to make their own arrangements to obtain the necessary documents.

If you have any questions about the application process not answered by the list of "Frequently Asked Questions" (below), please contact Mimi Hao, at gensci@fields.utoronto.ca

To be notified of future competitions, please subscribe to the Fields mail list.

 

2015 Research Problems

Project 1: Non-convex selfdual lagrangians
Supervisor: Abbas Momeni (Carleton University)
Research group: Yichao Chen, Qian Li, Michael Reynolds

Abstract: Many problems in science and engineering can be recast as variational problems. However, it is known that not every equation admits the standard Euler-Lagrange variational structure. In recent years, considerable effort has been devoted to the development of a calculus of variations that would go beyond this standard variational structure and that could be applied to a large number of partial differential systems and evolution equations, many of which do not fit within the classical Euler-Lagrange framework. The birth of the theory of convex and non-convex self-duality has been the result of some of these studies. The Weighted energy- dissipation functionals is another novel approach in this direction. The objective of this project is to develop a calculus on Non-convex self-dual Hamiltonians such as multiplication, addition and convolution in such a way that they preserve non-convex self-duality. It allows one to deal with rigorous problems in calculus of variations and optimization. (read more here)

Project 2: The Heisenberg group and uncertainty principle in mathematical physics
Supervisor: Hadi Salmasian (University of Ottawa)
Research group: Recep Celebi, Kirk Hendricks, Matthew Jordan

Abstract: One of the most remarkable features of quantum mechanics is Heisenberg's Uncertainty Principle, which roughly states that there is a limit to the precision of simultaneous measurements that can be made in particle physics. The classical mathematical model to explain this principle is based on an algebraic structure which is called the Heisenberg group. It is a miracle that the Heisenberg group appears in connection with many areas of mathematics, including number theory, analysis, and the theory of special functions.

The goal of this project is to study various connections of the Heisenberg group (and Heisenberg algebras) to mathematical physics, but also to other areas of mathematics, such as representation theory and harmonic analysis. This project is most suitable for students with a strong background in alge- braic structures (groups, rings and fields) and undergraduate-level analysis.

Project 3: Topics in delay differential equations
Supervisor: Jianhong Wu (York University)
Research Group: Noah Gelwan, Connor Harris, Changhan He, Bingchen Shan

Abstract: A delay differential equation (DDE) describes the evolution of a system for which the change rate of the state variable depends on the system's current and historical status. The initial state must be specified on an (initial) interval and an appropriate phase space must be infinite dimensional. This infinite dimensional and functional analytic framework in the case of constant delays was developed in the last century, and the qualitative study of DDEs has contributed to the advance in nonlinear analysis and infinite dimensional dynamical systems. In many applications arising from life sciences and engineering, the time delay is observed to depend on the system status.

This project of the 2015 Fields-Mitacs Undergraduate Summer Research Program will present an introduction of Delay Differential Equations in a level suitable for talented senior undergraduate students. Research topics of current interest, to be incorporated into the introduction, will include the study of the differences in qualitative behaviors of solutions to delay differential systems with different formats of state-dependence, and the study how to choose a format of state-dependence of delay to ensure "optimal" dynamic outcomes of the system under consideration. Specific examples from population dynamics and information processing will be used to motivate the study and demonstrate the theory.

Key references:
 1. The revised short introduction courses material, available 2nd week of June 2015.
 2. Thomas Erneux, Applied Delay Differential Equations, Springer, 2009.
 3. Shangjiang Guo and Jianhong Wu, Bifurcation Theory of Functional Differential Equations, Springer, 2013 (Recommended just for references)

Project 4: Topics in Statistical Inference for Big Data
Supervisors: Ejaz Ahmed (Brock University) and Vladimir Vinogradov (Ohio University, on leave at the Fields Institute, University of Toronto and York University)
Research Group: José Ibarra Rodriguez, Nada Khater, Hwanwoo Kim, Vsevolod Ladtchenko, Ilan Morgenstern, Ke Tong, Hongjing Zhang

Abstract: Today, more and more issues are arising in natural science and engineering, economics and finance, health research and social sciences which are all related to big data. For instance, they emerge in genomics, medical imaging, artificial intelligence, particle physics, the studies of both high-frequency financial data and network data to list a few. More recently, the statistical inference which relies on the shrinkage and selection method for linear regression known as the lasso has become popular. Specifically, this approach involves the construction of confidence intervals, the development of significance tests, and assigning the corresponding p-values in the analysis of high-dimensional data.

Another important direction is to study the shrinkage estimation for improving the prediction accuracy. In many scenarios where the model may be mis-specified, or there are potentially many variables with weak effects, a direct application of standard regularization methods may not be too useful. This suggests a potential and very interesting research topic whose main idea is to try to develop an alternative approach by combining the already available classical shrinkage estimation strategies with the state-of-the-art high-dimensional modeling techniques.

There are many challenging and interesting open problems in this new and rapidly developing area of statistics. Some of them will be presented to participants in the due course.

Key references:
 1. Ahmed, S.E. (2014). Penalty, Shrinkage and Pretest Strategies: Variable Selection and Estimation. New York: Springer.
 2. Ahmed, S. Ejaz, ed. (2014). Perspectives on Big Data Analysis: Methodologies and Applications. Providence, RI: AMS , vol. 622.
 3. Ahmed, S. Ejaz, ed. (2015). Big and Complex Data Analysis: Statistical Methodologies and Applications. New York: Springer (to appear).

Project 5: Topics in Computer Algebra
Supervisors: Rob Corless (Western), George Labahn (Waterloo), Greg Reid (Western), Stephen Watt (Western).
Research Group: Maksym Chaudkhari, Zhixing Guo, Bohdan Kivva, Oleksandr Rudenko, Torin Viger

Abstract: The field of computer algebra encompasses topics in constructive algebra, finding (or proving the existence of) algorithms in commutative algebra, linear algebra, differential algebra, algebraic and semi-algebraic geometry. Researchers devise constructions to solve problems in these areas and are interested in their computational complexity and suitability for effective implementation in systems such as Maple, Mathematica, Sage or Magma.

This summer's program will include problems in three areas:

1. Algorithms for symbolic polynomials.
The area of effective algebraic geometry replete with efficient algorithms for numerous problems, e.g. polynomial factorization, cylindrical algebraic decomposition, computation of standard bases, and so on. The polynomial systems may have coefficients from a variety of rings, including rational functions in parameters. Users of computer algebra systems, however, often pose problems where parameters appear in the exponents of the "polynomials". Several algorithms exist for these polynomial-like objects with symbolic exponents and suitable coefficient rings, including factorization and functional decomposition. This topic will explore additional problems in this area, connecting new theory with practical implementation.

2. Experimental Mathematics: Eigenvalues of structured random matrices
We have recently made some interesting progress in exploring the eigenvalues of certain matrices possessing discrete structure. However, much yet remains to be explored -- indeed there are an infinite number of possible structures that are of interest. The participating student will use HPC to explore experimentally some of these, in order to discover patters that reveal interesting mathematical phenomena that we will then go to (hopefully) prove.

3. Geometry, Symbolic Computation SDP and nonlinear PDE
Recent exciting progress in what is now known as "uncertainty quantification" mixes ideas from geometry and optimization to produce new algorithms to solve previously impossible problems. This project is doing something similar:
the accepted student will bring some skill (or enthusiasm) to join a team that is exploring the implications of a new geometrical view leading to new algorithms for nonlinear PDE.

Frequently Asked Questions

I am planning to graduate this upcoming June and I was wondering if I am still eligible to participate in this program?

Yes, but preference is given to students going into their final year or earlier. 

Is there a GPA requirement for students to apply?

No, but students with higher GPA rank higher during the selection process.

Are students without prior research experience in a Mathematical discipline, but with experience in, for example, eligible for the Program?

Yes, we welcome students with experience in any area of mathematical sciences.

Can the references be of character in nature?

Letters should address the academic and research backgrounds of the applicant as much as possible, in addition to character references if deemed relevant for the program.

I am interested in this year's research problems, but I am a graduate or postdoctoral level student. Can I still apply?

This Program is intended for undergraduate level students: if you have already completed an undergraduate degree in mathematics, you are not eligible for the Program.